Systems and Methods for Error Correction and Decoding on Multi-Level Physical Media

ABSTRACT

Apparatus and methods for operating a flash device characterized by use of Lee distance based codes in a flash device so as to increase the number of errors that can be corrected for a given number of redundancy cells, compared with Hamming distance based codes.

REFERENCE TO CO-PENDING APPLICATIONS

Priority is claimed from the following co-pending applications: U.S.Provisional Application No. 61/006,385, filed Jan. 10, 2008 and entitled“A System for Error Correction Encoder and Decoder Using the Lee Metricand Adapted to Work on Multi-Level Physical Media”, U.S. ProvisionalApplication No. 61/064,995, filed Apr. 8, 2008 and entitled “Systems andMethods for Error Correction and Decoding on Multi-Level PhysicalMedia”, U.S. Provisional Application No. 60/996,948, filed Dec. 12, 2007and entitled “Low Power BCH/RS Decoding: a Low Power Chien-SearchImplementation”, U.S. Provisional Application No. 61/071,487, filed May1, 2008 and entitled “Chien-Search System Employing a Clock-GatingScheme to Save Power for Error Correction Decoder and otherApplications” and U.S. Provisional Application No. 61/071,468, filedApr. 30, 2008 and entitled “A Low Power Chien-Search Based BCH/RSRecoding System for Flash Memory, Mobile Communications Devices andOther Applications”.

Other co-pending applications include: U.S. Provisional Application No.60/960,207, filed Sep. 20, 2007 and entitled “Systems and Methods forCoupling Detection in Flash Memory”, U.S. Provisional Application No.61/071,467, filed Apr. 30, 2008 and entitled “Improved Systems andMethods for Determining Logical Values of Coupled Flash Memory Cells”,U.S. Provisional Application No. 60/960,943, filed Oct. 22, 2007 andentitled “Systems and methods to reduce errors in Solid State Disks andLarge Flash Devices” and U.S. Provisional Application No. 61/071,469,filed Apr. 30, 2008 and entitled “Systems and Methods for AveragingError Rates in Non-Volatile Devices and Storage Systems”, U.S.Provisional Application No. 60/996,027, filed Oct. 25, 2007 and entitled“Systems and Methods for Coping with Variable Bit Error Rates in FlashDevices”, U.S. Provisional Application No. 61/071,466, filed Apr. 30,2008 and entitled “Systems and Methods for Multiple Coding Rates inFlash Devices”, U.S. Provisional Application No. 61/006,120, filed Dec.19, 2007 and entitled “Systems and Methods for Coping with Multi StageDecoding in Flash Devices”, U.S. Provisional Application No. 61/071,464,filed Apr. 30, 2008 and entitled “A Decoder Operative to Effect APlurality of Decoding Stages Upon Flash Memory Data and Methods Usefulin Conjunction Therewith”, U.S. Provisional Application No. 60/996,782,filed Dec. 5, 2007 and entitled “Systems and Methods for Using aTraining Sequence in Flash Memory”, U.S. Provisional Application No.61/064,853, filed Mar. 31, 2008 and entitled “Flash Memory Device withPhysical Cell Value Deterioration Accommodation and Methods Useful inConjunction Therewith”, U.S. Provisional Application No. 61/129,608,filed Jul. 8, 2008 and entitled “A Method for Acquiring and TrackingDetection Thresholds in Flash Devices”, U.S. Provisional Application No.61/006,806, filed Jan. 31, 2008 and entitled “Systems and Methods forusing a Erasure Coding in Flash memory”, U.S. Provisional ApplicationNo. 61/071,486, filed May 1, 2008 and entitled “Systems and Methods forHandling Immediate Data Errors in Flash Memory”, U.S. ProvisionalApplication No. 61/006,078, filed Dec. 18, 2007 and entitled “Systemsand Methods for Multi Rate Coding in Multi Level Flash Devices”, U.S.Provisional Application No. 61/064,923, filed Apr. 30, 2008 and entitled“Apparatus For Coding At A Plurality Of Rates In Multi-Level FlashMemory Systems, And Methods Useful In Conjunction Therewith”, U.S.Provisional Application No. 61/006,805, filed Jan. 31, 2008 and entitled“A Method for Extending the Life of Flash Devices”, U.S. ProvisionalApplication No. 61/071,465, filed Apr. 30, 2008 and entitled “Systemsand Methods for Temporarily Retiring Memory Portions”, U.S. ProvisionalApplication No. 61/064,760, filed Mar. 25, 2008 and entitled “Hardwareefficient implementation of rounding in fixed-point arithmetic”, U.S.Provisional Application No. 61/071,404, filed Apr. 28, 2008 and entitled“Apparatus and Methods for Hardware-Efficient Unbiased Rounding”, U.S.Provisional Application No. 61/136,234, filed Aug. 20, 2008 and entitled“A Method Of Reprogramming A Non-Volatile Memory Device WithoutPerforming An Erase Operation”, U.S. Provisional Application No.61/129,414, filed Jun. 25, 2008 and entitled “Improved Programming Speedin Flash Devices Using Adaptive Programming”, and several otherco-pending patent applications being filed concurrently (same day).

FIELD OF THE INVENTION

The present invention relates generally to error correction encoding anddecoding and more particularly to encoding and decoding in flash memorysystems.

BACKGROUND OF THE INVENTION

Many types of flash memory are known. Conventional flash memorytechnology is described in the following publications inter alia:

-   Paulo Cappelletti, Clara Golla, Piero Olivo, Enrico Zanoni, “Flash    Memories”, Kluwer Academic Publishers, 1999-   G. Campardo, R. Micheloni, D. Novosel, “CLSI-Design of Non-Volatile    Memories”, Springer Berlin Heidelberg New York, 2005

The Lee metric is a known metric-based error correction encoder/decoderfunctionality.

Prior art technologies related to the present invention include:

-   [1] “Error Correction Coding Mathematical Methods and Algorithms”,    Todd K. Moon, A JOHN WILEY & SONS, INC., 2005.-   [2] “Introduction to Coding Theory”, Ron M. Roth, Cambridge    University Press, 2006.-   [3] “Algebraic Codes for Data Transmission”, Richard E. Blahut,    Cambridge University Press, 2003.

References to square-bracketed numbers in the specification refer to theabove documents.

The disclosures of all publications and patent documents mentioned inthe specification, and of the publications and patent documents citedtherein directly or indirectly, are hereby incorporated by reference.

SUMMARY OF THE INVENTION

The following terms may be construed either in accordance with anydefinition thereof appearing in the prior art literature or inaccordance with the specification, or as follows:

Bit error rate (BER): a parameter that a flash memory devicemanufacturer commits to vis a vis its customers, expressing the maximumproportion of wrongly read bits (wrongly read bits/total number of bits)that users of the flash memory device need to expect at any time duringthe stipulated lifetime of the flash memory device e.g. 10 years.Block: a set of flash memory device cells which must, due to physicallimitations of the flash memory device, be erased together. Also termederase sector, erase block.Cell: A component of flash memory that stores one bit of information (insingle-level cell devices) or n bits of information (in a multi-leveldevice having 2 exp n levels). Typically, each cell comprises afloating-gate transistor. n may or may not be an integer. “Multi-level”means that the physical levels in the cell are, to an acceptable levelof certainty, statistically partionable into multiple distinguishableregions, plus a region corresponding to zero, such that digital valueseach comprising multiple bits can be represented by the cell. Incontrast, in single-level cells, the physical levels in the cell areassumed to be statistically partitionable into only two regions, onecorresponding to zero and one other, non-zero region, such that only onebit can be represented by a single-level cell.Charge level: the measured voltage of a cell which reflects its electriccharge.Cycling: Repeatedly writing new data into flash memory cells andrepeatedly erasing the cells between each two writing operations.Decision regions: Regions extending between adjacent decision levels,e.g. if decision levels are 0, 2 and 4 volts respectively, the decisionregions are under 0 V, 0 V-2 V, 2V-4 V, and over 4 V.Demapping: basic cell-level reading function in which a digital n-tupleoriginally received from an outside application is derived from aphysical value representing a physical state in the cell having apredetermined correspondence to the digital n-tuple.Digital value or “logical value”: n-tuple of bits represented by a cellin flash memory an capable of generating 2 exp n distinguishable levelsof a typically continuous physical value such as charge, where n may ormay not be an integer.Erase cycle: The relatively slow process of erasing a block of cells(erase sector), each block typically comprising more than one page, or,in certain non-flash memory devices, of erasing a single cell or theduration of so doing. An advantage of erasing cells collectively inblocks as in flash memory, rather than individually, is enhancedprogramming speed: Many cells and typically even many pages of cells areerased in a single erase cycle.Erase-write cycle: The process of erasing a block of cells (erasesector), each block typically comprising a plurality of pages, andsubsequently writing new data into at least some of them. The terms“program” and “write” are used herein generally interchangeably.Flash memory: Non-volatile computer memory including cells that areerased block by block, each block typically comprising more than onepage, but are written into and read from, page by page. IncludesNOR-type flash memory, NAND-type flash memory, and PRAM, e.g. SamsungPRAM, inter glia, and flash memory devices with any suitable number oflevels per cell, such as but not limited to 2, 4, or 8.Mapping: basic cell-level writing function in which incoming digitaln-tuple is mapped to a program level by inducing a program level in thecell, having a predetermined correspondence to the incoming logicalvalue.Page: A portion, typically 512 or 2048 or 4096 bytes in size, of a flashmemory e.g. a NAND or NOR flash memory device. Writing can be performedpage by page, as opposed to erasing which can be performed only erasesector by erase sector. A few bytes, typically 16-32 for every 512 databytes are associated with each page (typically 16, 64 or 128 per page),for storage of error correction information. A typical block may include32 512-byte pages or 64 2048-byte pages.Precise read, soft read: Cell threshold voltages are read at a precision(number of bits) greater than the number of Mapping levels (2̂n). Theterms precise read or soft read are interchangeable. In contrast, in“hard read”, cell threshold voltages are read at a precision (number ofbits) smaller than the number of Mapping levels (2̂n where n=number ofbits per cell).Present level, Charge level: The amount of charge in the cell. TheAmount of charge currently existing in a cell, at the present time, asopposed to “program level”, the amount of charge originally induced inthe cell (i.e. at the end of programming)Program: same as “write”.Program level (programmed level, programming level): amount of chargeoriginally induced in a cell to represent a given logical value, asopposed to “present level”.Reprogrammability (Np): An aspect of flash memory quality. This istypically operationalized by a reprogrammability parameter, also termedherein “Np”, denoting the number of times that a flash memory can bere-programmed (number of erase-write cycles that the device canwithstand) before the level of errors is so high as to make anunacceptably high proportion of those errors irrecoverable given apredetermined amount of memory devoted to redundancy. Typicallyrecoverability is investigated following a conventional aging simulationprocess which simulates or approximates the data degradation effect thata predetermined time period e.g. a 10 year period has on the flashmemory device, in an attempt to accommodate for a period of up to 10years between writing of data in flash memory and reading of the datatherefrom.Resolution: Number of levels in each cell, which in turn determines thenumber of bits the cell can store; typically a cell with 2̂n levelsstores n bits. Low resolution (partitioning the window, W, of physicalvalues a cell can assume into a small rather than large number of levelsper cell) provides high reliability.Retention: of original physical levels induced in the cells; retentionis typically below 100% resulting in deterioration of original physicallevels into present levels.Retention time: The amount of time that data has been stored in a flashdevice, typically without, or substantially without, voltage having beensupplied to the flash device i.e. the time which elapses betweenprogramming of a page and reading of the same page.Symbol: Logical valueThreshold level: the voltage (e.g.) against which the charge level of acell is measured. For example, a cell may be said to store a particulardigital n-tuple D if the charge level or other physical level of thecell falls between two threshold values T.

Certain embodiments of the present invention seek to provide improvedmethods for error correction encoding and decoding, for flash memory andother applications.

Reed Solomon (RS) codes are common error correction codes (ECC) used invarious applications such as storage and probably among the mostprevalent codes today. These codes are particularly common when theoutput from the physical media is binary and there is no additionalinformation regarding the reliability of the bits. In some cases thesecodes are also used to handle the case of erasures—i.e. bits whose valueis unknown.

Many of these codes are systematic. That is, the code may be dividedinto two sections, the first containing the original data and the secondcontaining spare bits, containing redundant information which allows adecoder to reconstruct the original data if errors have occurred in thefirst and/or second sections. The number of errors that can be correctedis a function of the code length and of the length of the redundancywithin the code.

In RS codes, the data sequence is a bit sequence which is divided intosubsets called symbols. The code itself is a sequence of symbols andonly symbols are corrected. It is irrelevant which bit or if severalbits (say f, the number of bits in a symbol) were in error in a symbol;the symbol is corrected as a unit. The code t has a limit on the numberof symbols that can be corrected regardless of how many bits werecorrected. That is, an RS code can correct all received sequences aslong as their Hamming distance from the original codeword is not largerthan a certain value given by D/2, where “Hamming distance” between twosequences of symbols is the count of a number of different symbolsbetween the sequences. D is also equal to the number of symbols in theredundancy section of the code; therefore, the redundancy would have D*fbits. A special case of RS codes are binary BCH codes, also known asalternant codes. These codes may be viewed as a subset of RS codes whereonly those code-words whose symbols are only 0 and 1 are used. Withthese codes it is possible to correct up to D errors with a redundancyof D*f bits. f is now chosen such that the overall length of the code(n) is smaller than n<2f. The code corrects sequences as long as theirHamming distance is less than or equal to D, where the Hamming distanceis taken over bits.

Certain embodiments of the present invention seek to provide a codewhich outperforms the binary BCH code for certain applications e.g.those in which short bit sequences are mapped to different levels—suchas, for example, in multi-level Flash memory devices where every 2-4bits are mapped to 4-16 levels.

Prior art FIG. 1 illustrates the distribution of the charge level in a 2bit/cell Flash device. Every 2 bits are mapped into one of the 4 levelsin each cell. Errors may occur if one charge level is read as another.Almost all errors occur when one level is mistaken for an adjacentlevel. Only rarely do errors occur between non-adjacent levels.

If a binary BCH code is used, gray mapping of the levels into bitsassists in reducing the number of bit error per programmed cell.However, the binary BCH code does not take into account the fact thatmore than one bit error per cell occurs only very rarely indeed.

There is thus provided, in accordance with at least one embodiment ofthe present invention, a method for error correction encoding of L levelapplication data residing in a memory comprising L level Multi-levelcells (MLCs) including at least some Multi-level cells (MLCs) in whichthe application data is residing and at least some Multi-level cells(MLCs) which are at least partly available to accept data other than theapplication data, the method comprising encoding the L level applicationdata over a prime field thereby to generate non-binary redundancy data,binarizing at least some of the non-binary redundancy data thereby togenerate binarized redundancy data, effecting binary error-correctionencoding of the binarized redundancy data, thereby to generate binaryredundancy data, combining the binarized redundancy data and the binaryredundancy data thereby to generate combined binarized/binary redundancydata; and restoring the combined binarized/binary redundancy data to Llevel form, thereby to generate restored L level redundancy data, andstoring the restored L level redundancy data in at least some of the atleast partly available L level Multi-level cells (MLCs).

Further in accordance with at least one embodiment of the presentinvention, the method also comprises, for at least one L level MLC,utilizing less than L levels for storing at least a portion of the Llevel application data and utilizing remaining ones of the L levels forstoring at least a portion of the combined binarized/binary redundancydata.

Still further in accordance with at least one embodiment of the presentinvention, binarizing comprises binarizing all of the non-binaryredundancy data.

Additionally in accordance with at least one embodiment of the presentinvention, binarizing comprises binarizing only some of the non-binaryredundancy data thereby to define a portion of the non-binary redundancydata which is un-binarized, and wherein the method also comprisesL-level transforming the un-binarized non-binary redundancy data toL-level un-binarized non-binary redundancy data.

Still further in accordance with at least one embodiment of the presentinvention, binarizing and L-level transforming are effected by a singletransform function.

Further in accordance with at least one embodiment of the presentinvention, the method also comprises storing the L-level un-binarizednon-binary redundancy data in the L level Multi-level cells (MLCs).

Still further in accordance with at least one embodiment of the presentinvention, the encoding over a prime field comprises Reed-Solomonencoding over a prime field.

Still further in accordance with at least one embodiment of the presentinvention, the combining comprises concatenating the binarizedredundancy data and the binary redundancy data thereby to generateconcatenated binarized/binary redundancy data.

Additionally in accordance with at least one embodiment of the presentinvention, the binary error-correction encoding comprises BCH encodingof the binarized redundancy data, thereby to generate BCH-encodedredundancy data.

Further in accordance with at least one embodiment of the presentinvention, the at least some Multi-level cells (MLCs) which are at leastpartly available to accept data other than the application data compriseentirely available Multi-level cells (MLCs) in which no L levelapplication data is residing, and wherein the storing comprises storingthe restored L level redundancy data in at least some of the entirelyavailable L level Multi-level cells (MLCs).

Also provided, in accordance with at least one embodiment of the presentinvention, is a method for error correction decoding of at leastrestored L level redundancy data residing in a memory comprising L levelMulti-level cells (MLCs) including at least some Multi-level cells(MLCs) in which possibly erroneous L-level application data resides andat least some Multi-level cells (MLCs) which store at least the restoredL level redundancy data, the method comprising binarizing the at leastrestored L level redundancy data, thereby to generate binarized restoredL level redundancy data, deriving binary redundancy data from thebinarized restored L level redundancy data, effecting binaryerror-correction decoding of the binary redundancy data, thereby togenerate binarized redundancy data, transforming the binarizedredundancy data to generate non-binary redundancy data, and decoding thenon-binary redundancy data and the possibly erroneous L-levelapplication data over a prime field, thereby to generate decoded L levelapplication data.

Further in accordance with at least one embodiment of the presentinvention, binarizing comprises binarizing only some of the non-binaryredundancy data thereby to define a portion of the non-binary redundancydata which is un-binarized, and wherein the method also comprisesL-level transforming the un-binarized non-binary redundancy data toL-level un-binarized non-binary redundancy data.

Also provided, in accordance with at least one embodiment of the presentinvention, is an apparatus for error correction encoding of L levelapplication data residing in a memory comprising L level Multi-levelcells (MLCs) including at least some Multi-level cells (MLCs) in whichthe application data is residing and at least some Multi-level cells(MLCs) which are at least partly available to accept data other than theapplication data, the apparatus comprising prime-field encodingapparatus operative to encode the L level application data over a primefield thereby to generate non-binary redundancy data, binarizationapparatus operative to binarize at least some of the non-binaryredundancy data thereby to generate binarized redundancy data, binaryECC apparatus operative to effect binary error-correction encoding ofthe binarized redundancy data, thereby to generate binary redundancydata; and data combination apparatus operative to combine the binarizedredundancy data and the binary redundancy data thereby to generatecombined binarized/binary redundancy data, and L-level restorationapparatus operative to restore the combined binarized/binary redundancydata to L level form, thereby to generate restored L level redundancydata, and storing the restored L level redundancy data in at least someof the at least partly available L level Multi-level cells (MLCs).

Additionally provided, in accordance with at least one embodiment of thepresent invention, is a method for operating a flash device, the methodcomprising using Lee distance based codes in the flash device, therebyincreasing the number of errors that can be corrected for a given numberof redundancy cells compared with Hamming distance based codes.

Also provided, in accordance with at least one embodiment of the presentinvention, is a method for operating a flash device, the methodcomprising providing a Lee code associated with a first group ofsymbols; and coding data in the flash device using a second group ofsymbols, the second group being a sub group of the first group, so as toallow extending the number of errors that can be corrected for a givennumber of redundancy cells compared with Hamming distance based codes.

Further provided, in accordance with yet another embodiment of thepresent invention, is a method for operating a flash device, the methodcomprising providing a Lee code associated with a first group ofsymbols, coding data in the flash device using a second group ofsymbols, the second group being a sub group of the first group, givingrise to a first data stream, producing a redundancy data stream from thefirst data stream using a redundancy code, thereby increasing the numberof errors that can be corrected for a given number of redundancy cellscompared with Hamming distance based codes. Further in accordance withat least one embodiment of the present invention, the first data streamis associated with a first group of symbols and the redundancy datastream is associated with a second group of symbols, the second groupbeing a sub group of the first group.

Further provided, in accordance with at least one embodiment of thepresent invention, is a decoding system comprising a Lee metric-baseddecoder operative to decode multi-level physical media.

Also provided, in accordance with at least one embodiment of the presentinvention, is a method for error correction comprising encoding anddecoding, thereby to effect error correction and using a separateencoder to store a Syndrome computation to be used during the decoding.

Yet further provided, in accordance with at least one embodiment of thepresent invention, is a method for error correction comprising encodingand decoding, thereby to effect error correction, including encoding asyndrome; and encapsulating and mapping the encoded syndrome intodesignated media cells.

Also provided, in accordance with at least one embodiment of the presentinvention, is a method for error correction comprising encoding anddecoding, thereby to effect error correction, including encoding asyndrome, and reducing error probability in the encoded Syndrome byencapsulating and mapping the encoded Syndrome mixed with data bits intodesignated media cells.

Also provided, in accordance with at least one embodiment of the presentinvention, is a method for error correction comprising encoding anddecoding, thereby to effect error correction, including multiplying byan inverse matrix to achieve a reduced number of bits sufficient toencode the syndrome and encoding the syndrome using only the reducednumber of bits.

Further in accordance with at least one embodiment of the presentinvention, the multi-level physical media includes media having lesslevels than symbol values.

Any suitable processor, display and input means may be used to process,display, store and accept information, including computer programs, inaccordance with some or all of the teachings of the present invention,such as but not limited to a conventional personal computer processor,workstation or other programmable device or computer or electroniccomputing device, either general-purpose or specifically constructed,for processing; a display screen and/or printer and/or speaker fordisplaying; machine-readable memory such as optical disks, CDROMs,magnetic-optical discs or other discs; RAMs, ROMs, EPROMs, EEPROMs,magnetic or optical or other cards, for storing, and keyboard or mousefor accepting. The term “process” as used above is intended to includeany type of computation or manipulation or transformation of datarepresented as physical, e.g. electronic, phenomena which may occur orreside e.g. within registers and/or memories of a computer.

The above devices may communicate via any conventional wired or wirelessdigital communication means, e.g. via a wired or cellular telephonenetwork or a computer network such as the Internet.

The apparatus of the present invention may include, according to certainembodiments of the invention, machine readable memory containing orotherwise storing a program of instructions which, when executed by themachine, implements some or all of the apparatus, methods, features andfunctionalities of the invention shown and described herein.Alternatively or in addition, the apparatus of the present invention mayinclude, according to certain embodiments of the invention, a program asabove which may be written in any conventional programming language, andoptionally a machine for executing the program such as but not limitedto a general purpose computer which may optionally be configured oractivated in accordance with the teachings of the present invention.

Any trademark occurring in the text or drawings is the property of itsowner and occurs herein merely to explain or illustrate one example ofhow an embodiment of the invention may be implemented.

Unless specifically stated otherwise, as apparent from the followingdiscussions, it is appreciated that throughout the specificationdiscussions, utilizing terms such as, “processing”, “computing”,“estimating”, “selecting”, “ranking”, “grading”, “calculating”,“determining”, “generating”, “reassessing”, “classifying”, “generating”,“producing”, “stereo-matching”, “registering”, “detecting”,“associating”, “superimposing”, “obtaining” or the like, refer to theaction and/or processes of a computer or computing system, or processoror similar electronic computing device, that manipulate and/or transformdata represented as physical, such as electronic, quantities within thecomputing system's registers and/or memories, into other data similarlyrepresented as physical quantities within the computing system'smemories, registers or other such information storage, transmission ordisplay devices.

BRIEF DESCRIPTION OF THE DRAWINGS

Certain embodiments of the present invention are illustrated in thefollowing drawings:

FIG. 1 is a prior art graph of physical level distributions in a two-bitper cell flash memory device;

FIG. 2 is a simplified functional block diagram of a “syndrome separate”encoder constructed and operative in accordance with certain embodimentsof the present invention;

FIG. 3 is a simplified functional block diagram of a “syndrome separate”decoder constructed and operative in accordance with certain embodimentsof the present invention;

FIG. 4 is a simplified functional block diagram of a “syndrome separate,redundancy & data mixed” encoder constructed and operative in accordancewith certain embodiments of the present invention;

FIG. 5 is a simplified functional block diagram of a “syndrome separate,redundancy & data mixed” decoder constructed and operative in accordancewith certain embodiments of the present invention;

FIG. 6 is a simplified functional block diagram of a “syndrome separate& shortened” encoder constructed and operative in accordance withcertain embodiments of the present invention;

FIG. 7 is a simplified functional block diagram of a “syndrome separate& shortened” decoder constructed and operative in accordance withcertain embodiments of the present invention;

FIG. 8 is a simplified functional block diagram of a “syndrome separate& shortened, redundancy & data mixed” encoder constructed and operativein accordance with certain embodiments of the present invention;

FIG. 9 is a simplified functional block diagram of a “syndrome separate& shortened, redundancy & data mixed” decoder constructed and operativein accordance with certain embodiments of the present invention;

FIG. 10A is a table representing a first mapping scheme in which eachsubsequence is mapped into a number represented by the binaryrepresentation of that sequence;

FIG. 10B is a table representing a second mapping scheme in which eachsubsequence is mapped into a symbol using “Gray coding”;

FIG. 11 is a simplified flowchart illustration of a method forencapsulating input data bits, in accordance with certain embodiments ofthe present invention;

FIG. 12 is a simplified flowchart illustration of a method forde-encapsulation which may be performed by Demapper B in the decoder ofFIG. 3, in accordance with certain embodiments of the present invention;

FIG. 13 is a simplified flowchart illustration of a method forde-encapsulation which may be performed by the mapper in the decoder ofFIG. 3, in accordance with certain embodiments of the present invention;

FIG. 14 is a diagram of a codeword generated by the encoder of FIG. 8,in accordance with certain embodiments of the present invention;

FIG. 15 is a diagram of a codeword generated by the encoder of FIG. 2,in accordance with certain embodiments of the present invention;

FIG. 16 is a diagram of a codeword generated by the encoder of FIG. 4,in accordance with certain embodiments of the present invention;

FIG. 17 is a diagram of a codeword generated by the encoder of FIG. 6,in accordance with certain embodiments of the present invention;

FIG. 18A is a simplified functional block diagram illustration of a NANDFlash device interacting with an external controller with Lee-metricbased error correction encoding & decoding functionality in accordancewith certain embodiments of the present invention;

FIG. 18B is a simplified functional block diagram illustration of aflash memory system including an on-board microcontroller, withLee-metric based error correction encoding & decoding functionality inaccordance with certain embodiments of the present invention, whereinthe encoding functionality may comprise any of the encoders of FIGS. 2,4, 6 and 8 and the decoding functionality may comprise any of thedecoders of FIGS. 3, 5, 7 and 9 respectively;

FIG. 19 is a simplified flowchart illustration of a method for effectinga NAND flash program command issued by the controller of FIG. 18A, inaccordance with certain embodiments of the present invention; and

FIG. 20 is a simplified flowchart illustration of a method for effectinga NAND flash read command issued by the controller of FIG. 18A, inaccordance with certain embodiments of the present invention.

DETAILED DESCRIPTION OF CERTAIN EMBODIMENTS

Apparatus and methods for operating a flash device are now described,which are characterized by use of Lee distance based codes in the flashdevice so as to increase the number of errors that can be corrected fora given number of redundancy cells, compared with Hamming distance basedcodes.

The Lee distance between two sequences of symbols is the absolute sum ofdifferences. If there are two symbols, S1 and S2, which take valuesbetween 0 and p−1 where p is a prime number, then the Lee distancebetween these two symbols is then given by

${D\left( {S_{1},S_{2}} \right)} = \left\{ {\begin{matrix}{{S_{1} - S_{2}}} & {{{S_{1} - S_{2}}} < {\left( {p - 1} \right)/2}} \\{p - {{S_{1} - S_{2}}}} & {{{S_{1} - S_{2}}} \geq {\left( {p - 1} \right)/2}}\end{matrix}.} \right.$

The distance between two sequences Si and Ri is then given by

$\sum\limits_{i = 0}^{n - 1}{D\left( {S_{i},R_{i}} \right)}$

where n is the number of symbols in each of the two sequences.

Thus, using the above distance, the measure between two sequences is thenumber of level shifts assuming that per symbol there is a maximum ofone error, i.e. either a single positive shift or a single negativeshift; the Lee distance measure is suitable since the fact that multipleshifts per symbol are rare is now taken into account. Generallyspeaking, gain, compared with binary BCH, results from the fact thatless bits are now used to store a full symbol.

In the described embodiments of the present invention, the use of theterms flash memory, flash memory device, flash memory apparatus, solidstate disk, memory device and similar is non-limiting andinterchangeable (also referred to in general as “storage apparatus”). Incertain embodiments of the present invention, the elements associatedwith the storage apparatus as well as the sequence of operations carriedout, are integral to the storage apparatus. In certain other embodimentsof the present invention, at least some of the elements associated withthe storage apparatus and at least some of the related operations areexternal to the flash, the solid state disk etc. For example, some ofthe operations performed by microcontroller 110 of FIG. 18A may becarried out by the central processing unit of the host 100, withoutlimiting the scope of the present invention. For convenience only, thefollowing description of certain embodiments of the present inventionrefers mainly to the term flash memory or solid state disk, howeverthose versed in the art will readily appreciate that the embodimentsthat are described with reference to flash memory or solid state diskare likewise applicable mutatis mutandis to other storage apparatuses.

Although the applicability of the current invention is not limited toFlash memory devices, these are considered throughout for the sake ofbrevity.

One application of this invention, as shown in FIG. 18A, is in NANDFlash applications such as SD cards and USB drives. In theseapplications, the system comprises a controller 2400 and at least oneNAND flash 2405 and perhaps many such as shown. The controller 2400interfaces to a host through a specified protocol. For example, in thecase of a USB drive the protocol is that defined by universal serial busprotocol. The controller 2400 then translates commands from the host.For example, read/write commands are translated into page read, pageprogram and block erase commands on the Flash device 2405 through theNAND interface protocol. In order to combat errors which may occur inthe multi-level NAND flash devices, code operative in accordance withcertain embodiments of this invention may be employed, e.g. by adding anencoding operation to each program command and adding a decodingoperation to each read command. The encoding/decoding operations shownand described herein may replace existing encode/decode operations usingless efficient codes.

The NAND devices 2405 are programmed page by page where each page maycontain cells that store data and cells that store redundancy. Both setsof cells are used, such that the data and redundancy are spread overboth. In the course of a NAND Flash interface program and read commandsthe page data is transferred as a sequence of bits and the bits are thenmapped into cells. During a program or read command issued by thecontroller 2400, the output of encoder 2320 is mapped into bits suchthat when stored in the Flash device, the bits are mapped back intosymbols in Flash cells as defined by the encoders' output. This processis shown in FIG. 19, for a programming procedure, and in FIG. 20, for aread command.

Taking a 4-level multi-level cell (MLC) device as an example, it iscommon to match each physical page (i.e. page in the Flash array 2420)into two logical pages such that each cell contains a bit from each ofthe logical pages, mapped using Gray coding. Here, the basic unit is twological pages and each NAND flash read/program command is a sequence oftwo read/program commands of a logical page.

Alternatively, as shown in FIG. 18B, a flash memory device may beprovided whose internal or on-board microcontroller has Lee metric-basederror correction encoder/decoder functionality capable of effectivelycorrecting more errors than a binary BCH code for a given length ofredundancy. Four examples of encoding and decoding schemes in accordancewith certain embodiments of the present invention, are now described,which employ four example codes respectively, termed the “syndromeseparate” code, “syndrome separate, redundancy & data mixed” code, the“syndrome separate & shortened” code and the “syndrome separate &shortened, redundancy & data mixed” code respectively. Encoders anddecoders employing these codes respectively, each constructed andoperative in accordance with certain embodiments of the presentinvention are presented in FIGS. 2 and 3; 4 and 5; 6 and 7; 8 and 9respectively.

The codes are used to encode information over a multi-level physicalmedium such as a Flash memory. Thus, the inputs and outputs of the Flashdevice are symbols containing L levels denoted by “0”, “1”, “2”, through“L−1”. The inputs to the encoder are always assumed to be a sequence ofbits while the inputs to the decoder are assumed to be a sequence ofFlash symbols. The flash symbols could be represented as a subset of afinite field. More precisely, these symbols may be viewed as a subset ofa finite-filed GF(p) where p is a prime number.

The encoders receive k bits, where k is a predetermined constant, andproduce n symbols, where n is a predetermined constant. n is such thatn*log₂L>k where n*log₂L−k is, generally speaking, the number of sparebits allocated for the code for error correction purposes.

The “syndrome separate” code is characterized by a separately encodedsyndrome. The “syndrome separate” code is now described with referenceto FIGS. 2 and 3, and using the following symbols:

k—number of data bits to be encoded

n—number of symbols in encoded word

L—number of levels

r—number of bits to be encapsulated by Mapper A (100 in FIG. 2)

t—number of symbols produced by Mapper A (100)

p—the smallest prime larger than t

D_(i) (i=0 . . . t−1)—output symbols of Mapper A (100) comprisingnumbers in GF(p) with values between 0 and L−1.

w—number of symbols produced by the Syndrome Computation block (110) ofFIG. 2.

S_(j) (j=0 . . . w−1)—output symbols of the Syndrome computation block(110) comprising numbers in GF(p).

α—a non-zero element of GF(p) which is also a primitive element inGF(p).

k′—the number of bits mapped by the Demapper unit (120) following themapping of the w symbols outputted by the Syndrome Computation block(110).

n′—number of bits at the output of the BCH encoder unit (130).

r′—number of symbols encapsulated at once by the Demapper unit (120).

FIG. 2 represents the “syndrome separate” encoder. The input to theencoder, a sequence of k bits, is mapped into t symbols by Mapper A(100). p is the smallest prime larger than or equal to t. If L is apower of 2, the data is divided into subsequences of log₂L bits, eachsubsequence being mapped into a symbol between 0 to L−1, these symbolsbeing numbers in the prime field GF(p). Mapping may be effected e.g. bymapping each subsequence into a number represented by the binaryrepresentation of that sequence as shown in the table of FIG. 10A.Another possibility is to use Gray coding as shown in the table of FIG.10B. Gray coding is not crucial for Mapper A because the performance ofthe code is not degraded if a different mapping is used, due to the factthat the encoding scheme shown and described herein works on entiresymbols and not on individual bits.

If L is not a power of 2 encapsulation may be used, typically dividingthe input sequence into subsequences of r bits which are then translatedinto a sequence of

$\left\lceil \frac{r}{\log_{2}L} \right\rceil$

symbols over GF(p) where each symbol is one of 0, 1, . . . , L−1. Thismay be done by representing the r bits as a number (in the standardnumber field) and following the procedure described in FIG. 11 torecover the symbols. In the method of FIG. 11, Z is, initially, thenumber represented by the r bits, and the following computations arerepeated

$\left\lceil \frac{r}{\log_{2}L} \right\rceil$

times: Symbol_(counter)=remainder of the division of Z by L; andcomputation of Z as the quotient of the division of the previous Z by L.r is a predefined constant chosen to maximize the efficiency ofencapsulation but yet sufficiently small to allow adequateimplementation of the method in FIG. 11 either in hardware or software.

The output of Mapper A (100) then goes to a delay line block (150) andto a Syndrome computation block (110). The purpose of the delay line(150) is to delay the sequence of the t symbols produced by the Mapper Ablock (100) such that the first symbols produced by (140) followimmediately after the last symbol was outputted from delay line (150).

The Syndrome computation block (110) produces w elements in GF(p). Asdescribed above, the t symbols outputted from Mapper A block (100) aretreated like numbers in the field GF(p). The Syndrome computation block(110) then performs the following computation:

$S_{j} = {\sum\limits_{i = 0}^{t - 1}{D_{i}\alpha^{ji}}}$

where D_(i) (i=0 . . . t−1) are the output symbols of Mapper A (100),S_(j) (j=0 . . . w−1) are the output symbols of the Syndrome computationblock (110) and a is a number between 1 and p−1 which is also aprimitive element in GF(p). All multiplications and additions arecarried out in the GF(p) field (i.e. modulo p). Finite fields, primitiveelements, and other related aspects of the above computations aredescribed in documents [2] and [3].

The output of the Syndrome computation unit (110) then flows to theDemapper unit (120). The Demapper maps the w GF(p) symbols into k′ bits,again, typically by using encapsulation. That is, the w symbols aredivided into subsets each including r′ symbols, other than, possibly,one subset which has less than r′ symbols. Each subset is then mappedinto bits using the binary representation of the number (now in thestandard field):

$\sum\limits_{i = 0}^{r^{\prime} - 1}{S_{i}{p^{i}.}}$

This number is mapped into a sequence of ┌r′·log₂ p┐ bits.

The output of the Demapper (120) flows into a binary BCH encoder (130)which receives k′ bits and produces n′ bits which include, on top of theoriginal k′ bits, additional n′−k′ redundancy bits which are used forerror correction in the decoding process. The binary BCH encoder is astandard encoder as is known in the art such as any of these describedin references [1], [2] and [3] and designed to function overGF(2^(┌ log) ² ^(n′┐)).

The output of the binary BCH encoder (130) is then mapped into symbolMapper B (140) which receives n′ bits and maps them into n−t symbols.The mapping is performed similarly to that performed by Mapper A,typically using Gray coding. Again, if the number of levels is not apower of 2, encapsulation is effected as for Mapper A. If L is a powerof 2 then the sequence of n′ bits is simply divided into subsets oflog₂L bits. If the sequence does not exactly divide, the sequence iszero padded until it is divided by log₂L without a quotient. The subsetsof log₂L bits are then mapped using Gray coding as shown e.g. in thetable of FIG. 10B.

A Selector (160) enables the flow of either the mapped original datafrom the delay line or the mapped redundancy data from Mapper B (140).At first, typically the selector enables the delay line to flow out;once this has occurred, the output of Mapper B (140) flows out. Thedelay line is designed to delay the output of Mapper A (100) such thatthe output of Mapper B is synchronized to the end of the output of thedelay line. The end result is a codeword as shown in FIG. 15.

As shown, the codeword of FIG. 15 typically comprises a first portion(1500) which includes a mapping of the data sequence into L levels, anda second portion (1510) which is used as redundancy for the purpose oferror correction of the information in the first portion 1500. Secondportion 1510 is also mapped into L levels and also typically comprisestwo portions: a first portion (1520) including the mapping of thesyndrome (computed from portion 1500) mapped into bits and then mappedto L levels; and a second portion (1530) including the mapping to Llevels of the redundancy as computed by the BCH encoder of the portion(1520).

FIG. 3 shows a schematic view of the “syndrome separate” decoder. The“syndrome separate” decoder of FIG. 3 typically receives n symbols fromthe Flash device and produces k correct bits. The n received symbolsbelong to one of L levels and may contain errors causing the wrong levelto be read. The first t symbols are associated with actual data whereasthe last n−t symbols are associated with redundancy.

The first t symbols and the last n−t symbols may be sent to the decoderof FIG. 3 simultaneously and analyzed simultaneously. If all n symbolsare not sent simultaneously to the decoder, it is more efficient to sendthe last n−t symbols, which store the redundancy, first, followed by thefirst t symbols. The redundancy may then be decoded as the rest of thesymbols are being sent to the decoder.

The first t symbols are sent to the Syndrome Computation unit whichtypically performs the same task as was performed by syndromecomputation unit 110 in the “syndrome separate” encoder of FIG. 2. Thelast n−t symbols are sent to a Demapper B 200 which performs theconverse of the operation performed by the Mapper B (140) in FIG. 2. TheDemapper receives n−t symbols and produces n′ bits.

If the number of levels, L, is a power of 2, the Demapper 200 simplyproduces log₂L bits for every symbol, the bits being produced by aprocess of Gray coding e.g. according to the table of FIG. 10B. If L isnot a power of 2, de-encapsulation is performed to reverse the effect ofencapsulation. That is, the n−t symbols are divided into sub-sequencesof ┌r/log₂ L┐ symbols from which the Demapper B 200 produces r bits.FIG. 12 shows a procedure which may be employed to perform theDe-encapsulation. In the method of FIG. 12, Symbol_(0 . . . ┌r/log) ₂_(L┐) is a subsequence of symbols and Z is initially 0.Z=Z+Symbol_(counter)*L^((Counter-1)) is repeated

$\left\lceil \frac{r}{\log_{2}L} \right\rceil$

times and the output is the binary representation of the final value ofZ.

The binary output of the Demapper 200 is a sequence of n′ bits which arepassed to the binary BCH decoder (210) which corrects any errors whichmight have occurred in the redundancy symbols. The BCH decoding may beeffected as known in the art e.g. as described in references ([1], [2],[3]).

The BCH decoder (210) outputs k′ bits which were used to code theSyndrome during the encoding process performed by the encoder of FIG. 2.The k′ bits are passed to a mapper unit (220) which produces w symbolsover GF(p). This time the symbols may assume values from 0 to p−1. Themapper unit (220) performs the converse of the operation performed bythe Demapper unit (120) in the encoder of FIG. 2, e.g. byDe-encapsulation. The k′ bits are divided into subsets of ┌r′·log₂ p┐.The symbols may be derived by representing the subset as a number (inthe normal field) and performing the method in FIG. 13. In the method ofFIG. 13, Z, initially, is the number represented by the ┌r′·log₂p┐ bitsand the following computations are repeated r′ times:Symbol_(counter)=remainder of the division of Z by p; and new Z=thequotient of the division of the current value of Z, by p.

The output of Mapper (220) is the accurate Syndrome whereas the outputof the Syndrome Computation unit (230) is the estimated syndromeaccording to the received symbols from the device. The differencebetween them is the syndrome only due to the error. Thus, the output ofthe subtract block (240) is the difference (symbol by symbol) of the twosyndromes and is the syndrome of the errors that occurred in the first tsymbols. Using the output of the subtraction unit 240, the eligible wordwhich is closest (by the Lee metric) to the word actually received, isrecovered by best eligible word recovery block 250. Any suitableconventional method can be used to effect the recovery, such as thatdescribed in steps 2-5 on page 311 Section 10.4 (“Decoding alternantcodes in the Lee metric”), in reference [2]. Once the correct symbolshave been recovered, Demapper A (260) maps the t symbols back into kbits. The Demapper A undoes the operation of Mapper A (100) in the“syndrome separate” encoder of FIG. 2, similarly to Demapper B. However,it is not necessary to use Gray mapping.

Parameter determination according to certain embodiments of the presentinvention is now described. The length of the input sequence (in bits)and the length of the code word (in symbols) are given by conventionaldesign preferences that are typically unaffected by the teachings of theinvention shown and described herein, so k and n are assumed to beknown. Therefore, values for the constant p and the number of symbolsthat contain data t are also immediately available. The encapsulationparameters r and r′ are determined by hardware complexity restrictions;for example, r may not be limited to less than 128 bits. The remainingparameters may be determined by trading off the number of BCH redundancybits n′−k′ and the number of Syndrome symbols w, it being appreciatedthat n′ bits may be stored in n−t symbols. The tradeoff is typicallybetween the number of bits (n′−k′) allocated to correct the “accurate”Syndrome and the number of symbols (w): allocated to correct the datareferred to above, as stored in the word referred to above.

P_(err) denotes the frame error rate and P_(cross) denotes theprobability that an individual level is erroneously read as either thefollowing or preceding level. The probability of error may beapproximated by assuming that errors only occur between nearby levels.

A decoding error may occur if either one of the following events occurs:

-   -   1. An error occurs in the decoding of “accurate” Syndrome (210).        i.e., there were more than └(n′−k′)/┌ log₂ n′┐┘ errors in the        (n-t) symbols allocated to store the redundancy.    -   2. An error occurs in the decoding of the data symbols using the        Lee metric decoder (240). i.e., there were w or more errors in        the first t symbols.        -   The probability of any of these events may be bounded from            above by

$P_{err} \leq {{\sum\limits_{i = {{\lfloor{{({n^{\prime} - k^{\prime}})}/{\lceil{\log_{2}n^{\prime}}\rceil}}\rfloor} + 1}}^{n - t}{\frac{\left( {n - t} \right)!}{{i!}{\left( {n - t - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{n - t - i}}} + {\sum\limits_{i = w}^{t}{\frac{t!}{{i!}{\left( {t - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{t - i}}}}$

-   -   -   Thus, by enumerating over w, a value may be selected which            reduces the above probability, typically to a minimum.

In “syndrome separate, redundancy & data mixed” code the redundancysymbols are no longer separated from the data symbols but rather mixedwith them. This is done in such a manner that the error probability inthe redundancy information is decreased. The downside is that p (theprime used for the Lee-metric code) increases with respect to n, whichmay decrease the error correction capability of the Lee-metric decoder.The number of levels, L, is assumed above to be a power of 2. However,where L is not a power of 2, encapsulation may be used, as describedgenerally above. The following symbols are used in the description ofthe “syndrome separate, redundancy & data mixed” code:

k—number of data bits to be encoded

n—number of symbols in encoded word

L—number of levels

t—number of symbols produced by Mapper A (300).

p—the smallest prime larger than n

D_(i) (i=0 . . . n−1)—Combined symbol output of Mapper A (300) and A′(310) comprising numbers in GF(p) with values between 0 and L−1.

w—number of symbols produced by the Syndrome Computation block (340)

S_(j) (j=0 . . . w−1)—output symbols of the Syndrome computation block(340) comprising numbers in GF(p).

k′—the number of bits mapped by the Demapper unit (350) following themapping of the w symbols outputted by the Syndrome Computation block(340).

n′—number of bits at the output of the BCH encoder unit (360).

b_(i)—bit i (=0 . . . n′−1) of the output sequence of the BCH encoderunit (360).

Y_(i)—the symbols (i=0 . . . n−1) at the output of the Mapper B (370)unit.

-   -   FIG. 4 shows a schematic view of an encoder for the “syndrome        separate, redundancy & data mixed” code. The input is a sequence        of bits (k bits) which flows into Mappers A (300) and A′ (310).        The first t·log₂ L bits are mapped by Mapper A into symbols over        GF(p) with values between 0 and L−1. Mapper divides the t·log₂ L        bits into subsequences of log₂ L which are mapped into symbols        using any type of mapping (for example the mappings in the        tables in FIG. 10). It is appreciated that the choice of the        mapping does not affect the performance of the code. The last        k−t·log₂ L bits of the input sequence are mapped by Mapper A′        into symbols over GF(p) with values between 0 and L/2-1. This is        done by dividing the remaining bits into subsequences of log₂        L−1 bits and mapping the subsequences into symbols according to        any mapping scheme.    -   The Selector (320) simply adjust the flow of the symbols such        that the symbols generated by Mapper A′ (310) follow the last        symbols generated by Mapper A (300). Overall, there are n        symbols (i.e. n=t+┌(k−t·log₂ L)/(log₂ L−1)┐).        The symbols flow into the syndrome computation unit (340) which        performs the same task as performed by the syndrome computation        unit (110) in the “syndrome separate” encoder of FIG. 2. This        unit produces w symbols over GF(p) (this time, the symbols take        any value between 0 and p−1).

The w symbols then flow into the Demapper (350) which performs the sametask as performed by the Demapper unit (120) in the “syndrome separate”encoder. The output of the Demapper (350) is k′ bits which flow into thebinary BCH encoder (360) which produces n′ bits. Again, unit 360 isidentical to unit 130 in the “syndrome separate” encoder of FIG. 2.

The output of the BCH encoder (360) then flows into Mapper B (370). Thepurpose of Mapper B is to combine the outputs of the Mapper A′ (310)with the output of the BCH encoder. More precisely, Mapper A′ (310)produces ┌(k−t·log₂ L)/(log₂ L−1)┐ symbols with values between 0 andL/2−1. The BCH encoder (360) produces n′ bits. Typically, n′=┌k−t·log₂L)/(log₂ L−1)┐. That is, t n-n′. The combined output is a symbol withvalues between 0 and L−1. This is done using the following formula:

Y _(i) =D _(i) +b _(i) ·L/2, i=n−n′ . . . n−1,

where Y_(i) are the symbols at the output of the Mapper B (370) unit,D_(i) are the symbols at the output of the Selector unit (320) and b_(i)are the bits (with values of 0 and 1) at the output of the BCH encoder(360).

The purpose of the Delay Line (330) is to delay the sequence of thesymbols such that symbols i=n−n′ . . . n−1 may enter Mapper B (370) whenthe output of the BCH encoder is ready. The selector unit 380 thenoutputs the first t symbols from the delay line and the rest of then′=n−t symbols from Mapper B.

The end result of this process is a codeword structured as shown in FIG.16 which comprises 2 portions. The first portion (2100) includes themapping of t·log₂ L data bits into L-level symbols. The second portion(2110) includes both data and redundancy mapped into L-level symbols.This is done by letting the data determine the log₂ L−1 LSB bits of theindex of the L-level symbol. The syndrome (after mapping it into bits)and the redundancy of the BCH codes determine the most significant bitof the index of the L-level symbol.

FIG. 5 gives a schematic view of a decoder for the “syndrome separate,redundancy & data mixed” code. The n symbols of the code flowsimultaneously into Demapper A′ (400) and Demapper A (430). Demapper A(430) handles separately the first t symbols and the latter n′ symbols.The first t symbols are passed ‘as is’ to the Syndrome computation unit(440) while the latter n′ symbols are stripped from the Redundancyinformation produced by the BCH encoder (360) in the encoder. This isdone as follows:

${\hat{F}}_{i} = \left\{ \begin{matrix}F_{i} & {i = {{0\mspace{14mu} \ldots \mspace{14mu} t} - 1}} \\{{rem}\left( {F_{i},2^{L - 1}} \right)} & {i = {t\mspace{14mu} \ldots \mspace{14mu} n}}\end{matrix} \right.$

where {circumflex over (F)}{circumflex over (F₁)} is the output ofDemapper A and F_(i) is the input. Demapper A′ (400) only handles thelast n′ symbols and recovers from them only the redundancy bits producedby the BCH encoder by computing b_(i)=└Y_(i)/(L/2)┘ for i=t . . . n−1.

The output of Demapper A′ (400) is a sequence of n′ bits which aredecoded by the binary BCH decoder (410) which produces k′ bits. Thesebits are then mapped back into w symbols over GF(p) by the Mapper (420)which performs the same task as the Mapper (220) in the “syndromeseparate” decoder of FIG. 3.

In parallel, the output of Demapper A (430) flows into the Syndromecomputation unit (440) which performs the same task as the Syndromecomputation unit (340) in the encoder. The output of Mapper (420) arethe “accurate” syndrome symbols while the output of the syndromecomputation unit (440) is a result of errors symbols. The differencebetween these syndromes (symbol by symbol) is the syndrome due to theerrors alone. This difference is computed by the Subtract unit (450).

Unit 460 receives the output of 450 and Demapper A, and recovers theaccurate symbols. Any suitable conventional method can be used to effectthe recovery, such as that described in steps 2-5 on page 311 Section10.4 (“Decoding alternant codes in the Lee metric”), in reference [2].The output of 460 flows into the Demapper B (470) which performs thereverse operation of Mapper A (300) for symbols 0 . . . t−1 and MapperA′ (310) for symbols t . . . n−1. The output is the reconstructed inputstream.

Parameter Determination in accordance with certain embodiments of thepresent invention is now described. There is a degree of freedom lying,again, in the tradeoff between w which determines the k′ and the abilityto correct the data symbols and n′−k′ which is the redundancy allottedfor the BCH code and determines the ability to correct errors in the“accurate” syndrome. A decoding error may occur if either one of thefollowing events occurs:

-   -   1. An error occurs in the decoding of “accurate” Syndrome (410).        i.e., there were more than └n′−k′)/┌ log₂ n′┐┘ errors in the        (n−t) symbols allocated to store the redundancy. The probability        of bit error is P_(cross)/(L/2).    -   2. An error occurs in the decoding of the data symbols using the        Lee metric decoder (460) indicating that there were w or more        errors in one of the n symbols.        -   The probability of any of these events may be upper-bounded            by

$P_{err} \leq {{\sum\limits_{i = {{\lfloor{{({n^{\prime} - k^{\prime}})}/{\lceil{\log_{2}n^{\prime}}\rceil}}\rfloor} + 1}}^{n^{\prime}}{\frac{n^{\prime}!}{{i!}{\left( {n^{\prime} - i} \right)!}}\left( {\frac{2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{2}{L}P_{cross}}} \right)^{n^{\prime} - i}}} + {\sum\limits_{i = w}^{n}{\frac{n!}{{i!}{\left( {n - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{n - i}}}}$

-   -   -   Thus, by enumerating over w, a value may be selected which            brings the above probability to a minimum.        -   The improvement in the above expression here compared with            that obtained for the “syndrome separate” code is in the            first summand which now includes a small probability:            P_(cross)(L/2). This probability may be compared with the            probability appearing for the “syndrome separate” code after            normalizing with the number of bits per symbol; the            comparison being

$2{P_{cross}/L}\mspace{14mu} {vis}\mspace{14mu} a\mspace{14mu} {vis}\mspace{14mu} 2{{P_{cross}/L} \cdot {\frac{L - 1}{\log_{2}L}.}}$

The “syndrome separate & shortened” code, characterized by a separatelyencoded shortened Syndrome, is now described. The “syndrome separate &shortened” code embodiment improves the “syndrome separate” codeembodiment at the expense of adding w² multiplications over GF(p). Thefollowing symbols are used to describe the “syndrome separate &shortened” code:

k—number of data bits to be encoded

n—number of symbols in encoded word

L—number of levels

r—number of bits to be encapsulated by Mapper A (500)

t—number of symbols produced by Mapper A (500)

p—the smallest prime larger than t+w

D_(i) (i=0 . . . t−1)—output symbols of Mapper A (500) comprisingnumbers in GF(p) with values between 0 and L−1.

w—number of symbols produced by the Syndrome Computation block (510)

S_(j) (j=0 . . . w−1)—output symbols of the Syndrome computation block(510) comprising numbers in GF(p).

Z_(j) (j=0 . . . w−1)—output symbols of block 520 comprising numbers inGF(p).

V_(j) (j=0 . . . w−1) and T_(j) (j=0 . . . w−1)—output symbols of block530 comprising numbers in GF(p).

α—a non-zero element of GF(p) (i.e. a number between 1 and p−1).

k′—the number of bits mapped by the Demapper unit (540) following themapping of the w symbols outputted from the Roundup unit (530).

n′—number of bits at the output of the BCH encoder unit (550).

r′—number of symbols encapsulated in unison (in parallel) by theDemapper unit (540).

FIG. 6 illustrates an encoder of the “syndrome separate & shortened”code which is similar to the “syndrome separate” encoder of FIG. 2 butincludes 2 additional units: xG⁻¹ unit 520 and roundup unit 530. Theoperations of all other blocks substantially resemble the operations oftheir counterparts in FIG. 2, respectively. For example, as in the“syndrome separate” code embodiment, data in the form of a bit sequenceflows into Mapper A (500) and then to the syndrome computation unit(510). Both units, 500 and 510, operate substantially as do theircounterparts in the “syndrome separate” code encoder of FIG. 2, howeverthe w-symbol output of the Syndrome computation unit (510) flows into amatrix multiplier xG⁻¹ (520) which multiplies the syndrome (as computedbefore) by the following matrix G⁻¹:

${G^{- 1} = \begin{pmatrix}\alpha^{t \cdot 0} & \alpha^{{({t + 1})} \cdot 0} & \alpha^{{({t + 2})} \cdot 0} & \ldots & \alpha^{{({t + w - 1})} \cdot 0} \\\alpha^{t \cdot 1} & \alpha^{{({t + 1})} \cdot 1} & \alpha^{{({t + 2})} \cdot 1} & \; & \; \\\alpha^{t \cdot 2} & \alpha^{{({t + 1})} \cdot 2} & \alpha^{{({t + 2})} \cdot 2} & \; & \; \\\vdots & \; & \; & \ddots & \; \\\alpha^{t \cdot {({w - 1})}} & \; & \; & \; & \alpha^{{({t + w - 1})} \cdot {({w - 1})}}\end{pmatrix}^{- 1}},$

where α is the primitive element in GF(p) used in the computation of thesyndrome.

The multiplication operation is defined as follows:

${\begin{pmatrix}Z_{0} \\Z_{1} \\\vdots \\Z_{w - 1}\end{pmatrix} = {G^{- 1}\begin{pmatrix}S_{0} \\S_{1} \\\vdots \\S_{w - 1}\end{pmatrix}}},$

where Z_(i) are the outputs of the matrix multiplication unit.

The output of the matrix multiplier flows into the Roundup unit 530which adds a number T_(i) between 0 and L−1 to each of the elementsZ_(i) such that V_(i)=Z_(i)+T₁ modulo p is divisible by L. The symbolsT_(i), which take only values between 0 and L−1, are outputted to theSelector unit (580) which appends these symbols to the end of the tsymbols holding the data. The symbols V_(i) flow to the Demapper unit(540). V_(i) typically only have ┌p/L┐ possible values, all divisible byL.

The Demapper unit (540) operates similarly to the Demapper (120) of the“syndrome separate” code encoder of FIG. 2, however before performingthe encapsulation, the symbols are divided by L and in the encapsulationprocess, the parameter p is exchanged with ┌p/L┐. This yields a morecompact presentation of the syndrome and diminishes the BCH code'srequirements, thus allowing a tradeoff between w and n′−k′ in thecomputation of the frame error rate to weigh in favor of w. Theoperations of the other units in FIG. 6 are similar to those of theircounterparts in the “syndrome separate” encoder of FIG. 2. The endresult is a codeword as shown in FIG. 17.

The codeword of FIG. 17 may include three portions. The first portion(2200) is simply a mapping of the data sequence into L levels. Thesecond portion (2210) includes W L-level symbols which define a portionof the syndrome characterized in that the overall syndrome of the datasymbols along with these W symbols has a significantly smaller set ofpossibilities. The third portion (2220) includes some of the encodedsyndrome (only the portion thereof which was to be defined) and is usedas redundancy for purpose of error correction of the information in thefirst two portions 2200 and 2210. The third portion 2220 is also mappedinto L levels and comprises:

a first portion (2230) including the mapping of the total syndrome(computed from portion 2200 and 2210) mapped into bits and then mappedto L levels; and

a second portion (2240) including the mapping to L levels of theredundancy as computed by the BCH encoder of the portion (2230).

FIG. 7 illustrates a “syndrome separate & shortened” decoder which issimilar to the “syndrome separate” decoder of FIG. 3, however, a matrixmultiplier unit xG (640) is provided which multiplies the W-symboloutput of the Mapper unit (630) by the following matrix G:

$G = {\begin{pmatrix}\alpha^{t \cdot 0} & \alpha^{{({t + 1})} \cdot 0} & \alpha^{{({t + 2})} \cdot 0} & \ldots & \alpha^{{({t + w - 1})} \cdot 0} \\\alpha^{t \cdot 1} & \alpha^{{({t + 1})} \cdot 1} & \alpha^{{({t + 2})} \cdot 1} & \; & \; \\\alpha^{t \cdot 2} & \alpha^{{({t + 1})} \cdot 2} & \alpha^{{({t + 2})} \cdot 2} & \; & \; \\\vdots & \; & \; & \ddots & \; \\\alpha^{t \cdot {({w - 1})}} & \; & \; & \; & \alpha^{{({t + w - 1})} \cdot {({w - 1})}}\end{pmatrix}.}$

The Mapper unit (630) is similar to Mapper unit (220) in the “syndromeseparate” decoder of FIG. 2, however, the parameter p is replaced with┌p/L┐ and the outputs are multiplied by L. Also, the syndromecomputation unit (610) operates over t+w elements whereas itscounterpart in the “syndrome separate” decoder of FIG. 2 operates over telements. The remaining units in the “syndrome separate & shortened”decoder of FIG. 7 can be generally identical to their counterparts inthe “syndrome separate” code decoder of FIG. 2.

Parameter Determination for the “syndrome separate & shortened” codeembodiment of FIGS. 6 and 7 in accordance with certain embodiments ofthe present invention is now described. Computations for the “syndromeseparate & shortened” code embodiment differ from those performed forthe “syndrome separate” code to accommodate for the fact that a portionof the redundancy is passed to the Lee decoder and that k′ and n′ arenow smaller. A decoding error may occur if either one of the followingoccurs:

-   -   1. An error occurs in the decoding of “accurate” Syndrome (610),        e.g. there are more than └(n′−k′)/┌ log₂ n′┐┘ errors in the        (n−t−w) symbols allocated to store the redundancy.    -   2. An error occurs in the decoding of the data symbols using the        Lee metric decoder (660), e.g. there are w or more errors in the        first t+w symbols.

The probability of either of these events may be bounded from above by

$P_{err} \leq {{\sum\limits_{i = {{\lfloor{{({n^{\prime} - k^{\prime}})}/{\lceil{\log_{2}n^{\prime}}\rceil}}\rfloor} + 1}}^{n - t - w}{\frac{\left( {n - t - w} \right)!}{{i!}{\left( {n - t - w - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{n - t - i}}} + {\sum\limits_{i = w}^{t + w}{\frac{\left( {t + w} \right)!}{{i!}{\left( {t + w - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{t + w - i}}}}$

Thus, by enumerating over w, a value may be selected which reduces theabove probability, typically to a minimum.

The “syndrome separate & shortened, redundancy & data mixed” code,characterized by mixed data and an encoded shortened Syndrome, is nowdescribed. This code improves the “syndrome separate, redundancy & datamixed” code at the cost of adding w² multiplications over GF(p). Thefollowing symbols are used in the following description of the syndromeseparate & shortened, redundancy & data mixed” code:

k—number of data bits to be encoded

n—number of symbols in encoded word

L—number of levels

t—number of symbols produced by Mapper A (700).

p—the smallest prime larger than n D_(i) (i=0 . . . n−1)—Combined symboloutput of Mapper A (700) and A′ (710) Comprising numbers in GF(p) withvalues between 0 and L−1.

w—number of symbols produced by the Syndrome Computation block (740)

S_(j) (j=0 . . . w−1)—output symbols of the Syndrome computation block(740) comprising numbers in GF(p).

Z_(j) (j=0 . . . w−1)—output symbols of block 750 comprising numbers inGF(p).

V_(j) (j=0 . . . w−1) and T_(j) (j=0 . . . w−1)—output symbols of block760 comprising numbers in GF(p).

k′—the number of bits mapped by the Demapper unit (780) following themapping of the w symbols outputted by the Syndrome Computation block(740).

n′—number of bits at the output of the BCH encoder unit (790).

b_(i)—bit i (=0 . . . n′−1) of the output sequence of the BCH encoderunit (790).

Y_(i)—the symbols (i=0 . . . n−1) at the output of the Mapper B (800)unit

FIG. 8 illustrates a “syndrome separate & shortened, redundancy & datamixed” encoder. A difference between this encoder and the encoder of the“syndrome separate, redundancy & data mixed” code is that w additionalsymbols with values between 0 and L−1 are appended at the encoder ofcodeword 2, allowing the sequence of bits (b_(i), i=0 . . . n′−1) usedto encode the Syndrome to be shortened at the expense of adding welements to the Lee decoding procedure. This guarantees a new tradeoffbetween w and the size of the redundancy of the BCH code e.g. (n′−k′bits). The composition of the codeword is shown in FIG. 14. Themid-section 1410 comprises both data bits and encoded syndrome bits(what is left of it following the use of the additional w symbolsappended to the end of the codeword) as shown in FIG. 16 in portions2120 and 2130.

Mapper A (700), Mapper B (710), Selector (720) and the syndromeComputation unit (740) may be generally identical to their counterpartsin the “syndrome separate, redundancy & data mixed” encoder of FIG. 4namely units 300, 310, 320 and 340 respectively. It is appreciated thatin this embodiment, t=n−n′−w.

Downstream of the syndrome computation unit (740) are a matrixmultiplier xG⁻¹ (750) and a roundup unit (760) which may be generallysimilar to their counterparts in the “syndrome separate & shortened”encoder of FIG. 6 namely units 520 and 530 respectively. The matrixmultiplier xG⁻¹ (750) typically computes the following:

$\begin{pmatrix}Z_{0} \\Z_{1} \\\vdots \\Z_{w - 1}\end{pmatrix} = {\begin{pmatrix}\alpha^{{({t + n^{\prime}})} \cdot 0} & \alpha^{{({t + n^{\prime} + 1})} \cdot 0} & \alpha^{{({t + n^{\prime} + 2})} \cdot 0} & \ldots & \alpha^{{({t + n^{\prime} + w - 1})} \cdot 0} \\\alpha^{{({t + n^{\prime}})} \cdot 1} & \alpha^{{({t + n^{\prime} + 1})} \cdot 1} & \alpha^{{({t + n^{\prime} + 2})} \cdot 1} & \; & \; \\\alpha^{{({t + n^{\prime}})} \cdot 2} & \alpha^{{({t + n^{\prime} + 1})} \cdot 2} & \alpha^{{({t + n^{\prime} + 2})} \cdot 2} & \; & \; \\\vdots & \; & \; & \ddots & \; \\\alpha^{{({t + n^{\prime}})} \cdot {({w - 1})}} & \; & \; & \; & \alpha^{{({t + n^{\prime} + w - 1})} \cdot {({w - 1})}}\end{pmatrix}^{- 1}\begin{pmatrix}S_{0} \\S_{1} \\\vdots \\S_{w - 1}\end{pmatrix}}$

where Z_(i) are the outputs of the matrix multiplication unit.

The Demapper unit (780) may be similar to Demapper (350) of the“syndrome separate, redundancy & data mixed” encoder of FIG. 4, howevertypically, before demapper 780 performs encapsulation, the symbols aredivided by L and in the encapsulation process, the parameter p isexchanged with ┌p/L┐, all in contrast to demapper 350, to provide a morecompact presentation of the syndrome and diminish the requirements fromthe BCH, thereby allowing the tradeoff between w and n′−k′ in thecomputation of the frame error rate to weigh in favor of w. Theoperation of the other units may be similar to that of theircounterparts in the “syndrome separate, redundancy & data mixed” encoderof FIG. 4 except that the Mapper 800, unlike its counterpart, mapper 370in the “syndrome separate, redundancy & data mixed” code embodiment,effects mixing of the bits b₁ from the BCH encoder 790 and the channelsymbols for symbols t . . . t+n′−1 instead of for symbols n−n′ . . . n−1where t=n−n′−w, rather than: t=n−n′.

FIG. 9 illustrates a “syndrome separate & shortened, redundancy & datamixed” decoder which is similar to the “syndrome separate, redundancy &data mixed” decoder of FIG. 5, however, in FIG. 9, a matrix multiplierunit xG (930) is provided which receives the output of w symbols of theMapper unit (920) and multiplies it by the following matrix G:

$G = {\begin{pmatrix}\alpha^{{({t + n^{\prime}})} \cdot 0} & \alpha^{{({t + n^{\prime} + 1})} \cdot 0} & \alpha^{{({t + n^{\prime} + 2})} \cdot 0} & \ldots & \alpha^{{({t + n^{\prime} + w - 1})} \cdot 0} \\\alpha^{{({t + n^{\prime}})} \cdot 1} & \alpha^{{({t + n^{\prime} + 1})} \cdot 1} & \alpha^{{({t + n^{\prime} + 2})} \cdot 1} & \; & \; \\\alpha^{{({t + n^{\prime}})} \cdot 2} & \alpha^{{({t + n^{\prime} + 1})} \cdot 2} & \alpha^{{({t + n^{\prime} + 2})} \cdot 2} & \; & \; \\\vdots & \; & \; & \ddots & \; \\\alpha^{{({t + n^{\prime}})} \cdot {({w - 1})}} & \; & \; & \; & \alpha^{{({t + n^{\prime} + w - 1})} \cdot {({w - 1})}}\end{pmatrix}.}$

Demappers A′ (900) and A (940) may be similar to Demappers A′ (400) andA (430) in the “syndrome separate, redundancy & data mixed” decoder ofFIG. 5, except that demixing is effected for symbols t . . . t+n′−1instead of for symbols n−n′ . . . n−1 where t=n−n′−w rather than t=n−n′.The Mapper unit (920) may be generally similar to the Mapper unit (420)in the “syndrome separate” decoder of FIG. 3 however duringde-encapsulation, the parameter p is replaced with ┌p/L┐ and the outputsare multiplied by L. In the “syndrome separate & shortened, redundancy &data mixed” decoder of FIG. 9, as opposed to its FIG. 5 counterpart forthe “syndrome separate, redundancy & data mixed” code, the syndromecomputation unit (950) operates over n=t+n′+w elements instead of telements. The remaining units of FIG. 9 may be generally similar totheir “syndrome separate, redundancy & data mixed” code counterparts inthe embodiment of FIG. 5.

Parameter Determination in accordance with certain embodiments of thepresent invention is now described. A decoding error may occur if eitherone of the following events occurs:

-   -   1. An error occurs in the decoding of “accurate” Syndrome (910),        e.g. there are more than └(n′−k′)/┌ log₂ n′┐┘ errors in the        (n′=n−t−w) symbols allocated to store the redundancy. Now, the        probability of bit error is given by P_(cross)/(L/2).    -   2. An error occurs in the decoding of the data symbols using the        Lee metric decoder (970), e.g. there were w or more errors in n        symbols.        -   The probability of any of these events may be bounded from            above by

$P_{err} \leq {{\sum\limits_{i = {{\lfloor{{({n^{\prime} - k^{\prime}})}/{\lceil{\log_{2}n^{\prime}}\rceil}}\rfloor} + 1}}^{n^{\prime}}{\frac{n^{\prime}!}{{i!}{\left( {n^{\prime} - i} \right)!}}\left( {\frac{2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{2}{L}P_{cross}}} \right)^{n^{\prime} - i}}} + {\sum\limits_{i = w}^{n}{\frac{n!}{{i!}{\left( {n - i} \right)!}}\left( {\frac{{2L} - 2}{L}p_{cross}} \right)^{i}\left( {1 - {\frac{{2L} - 2}{L}P_{cross}}} \right)^{n - i}}}}$

-   -   Thus, by enumerating over w, a value may be selected which        reduces the above probability, e.g. to a minimum.

A numerical example for use of “syndrome separate & shortened,redundancy & data mixed” code is now described. In the followingexample, the number of levels is L=16, the code length is n=4352, thenumber of data bits to be encoded is k=16384, the number of symbolsproduced by Mapper A is t=5462, the prime number of the field is p=4357.the primitive element is α=2, the syndrome size is w=76, the number ofbits encapsulated by mapper r is 122 (encapsulating 15 symbols with┌p/L┐ values each), the bit sequence to be encoded by the BCH is k′=619bits long, the Bit sequence produced by the BCH is n′=719 bits long, thefield over which the binary BCH was constructed is GF(2¹⁰) and the primepolynomial constructing it is X¹⁰+X³+1. A code with the above parameterscan handle p_(cross)=0.0036 and obtain a frame error rate ofapproximately 5e-13. The advantage of the above code, in comparison withother conventional coding schemes, is immediately apparent. The abovecross over probability induces an un-coded bit error rate of 0.001688 ifGray coding is used; and it is appreciated that the code above did notactually require Gray coding. A frame error of 5E-13 would be obtainedby an equivalent code which can correct up to 75 errors. However, usinga conventional binary BCH code with the same number of spare cells (1024bits) it is only possible to correct 68 errors.

Use of Lee distance based codes rather than Hamming distance based codesis a particular advantage in flash memory devices constructed andoperative in accordance with certain embodiments of the presentinvention because if gray mapping is being used in a flash memorydevice, errors almost always involve a symbol being confounded with aneighboring symbol in the mapping scheme rather than with a distantsymbol in the mapping scheme, such that use of Lee distance based codemay give better error correction functionality per unit of redundancy orless redundancy for a given level of error correction functionalityadequacy.

It is appreciated that software components of the present inventionincluding programs and data may, if desired, be implemented in ROM (readonly memory) form including CD-ROMs, EPROMs and EEPROMs, or may bestored in any other suitable computer-readable medium such as but notlimited to disks of various kinds, cards of various kinds and RAMs.Components described herein as software may, alternatively, beimplemented wholly or partly in hardware, if desired, using conventionaltechniques.

Included in the scope of the present invention, inter alia, areelectromagnetic signals carrying computer-readable instructions forperforming any or all of the steps of any of the methods shown anddescribed herein, in any suitable order; machine-readable instructionsfor performing any or all of the steps of any of the methods shown anddescribed herein, in any suitable order; program storage devicesreadable by machine, tangibly embodying a program of instructionsexecutable by the machine to perform any or all of the steps of any ofthe methods shown and described herein, in any suitable order; acomputer program product comprising a computer useable medium havingcomputer readable program code having embodied therein, and/or includingcomputer readable program code for performing, any or all of the stepsof any of the methods shown and described herein, in any suitable order;any technical effects brought about by any or all of the steps of any ofthe methods shown and described herein, when performed in any suitableorder; any suitable apparatus or device or combination of such,programmed to perform, alone or in combination, any or all of the stepsof any of the methods shown and described herein, in any suitable order;information storage devices or physical records, such as disks or harddrives, causing a computer or other device to be configured so as tocarry out any or all of the steps of any of the methods shown anddescribed herein, in any suitable order; a program pre-stored e.g. inmemory or on an information network such as the Internet, before orafter being downloaded, which embodies any or all of the steps of any ofthe methods shown and described herein, in any suitable order, and themethod of uploading or downloading such, and a system including server/sand/or client/s for using such; and hardware which performs any or allof the steps of any of the methods shown and described herein, in anysuitable order, either alone or in conjunction with software.

Certain operations are described herein as occurring in themicrocontroller internal to a flash memory device. Such description isintended to include operations which may be performed by hardware whichmay be associated with the microcontroller such as peripheral hardwareon a chip on which the microcontroller may reside. It is alsoappreciated that some or all of these operations, in any embodiment, mayalternatively be performed by the external, host-flash memory deviceinterface controller including operations which may be performed byhardware which may be associated with the interface controller such asperipheral hardware on a chip on which the interface controller mayreside. Finally it is appreciated that the internal and externalcontrollers may each physically reside on a single hardware device, oralternatively on several operatively associated hardware devices.

Any data described as being stored at a specific location in memory mayalternatively be stored elsewhere, in conjunction with an indication ofthe location in memory with which the data is associated. For example,instead of storing page- or erase-sector-specific information within aspecific page or erase sector, the same may be stored within the flashmemory device's internal microcontroller or within a microcontrollerinterfacing between the flash memory device and the host, and anindication may be stored of the specific page or erase sector associatedwith the cells.

It is appreciated that the teachings of the present invention can, forexample, be implemented by suitably modifying, or interfacing externallywith, flash controlling apparatus. The flash controlling apparatuscontrols a flash memory array and may comprise either a controllerexternal to the flash array or a microcontroller on-board the flasharray or otherwise incorporated therewithin. Examples of flash memoryarrays include Samsung's K9XXG08UXM series, Hynix' HY27UK08BGFM Series,Micron's MT29F64G08TAAWP or other arrays such as but not limited to NORor phase change memory. Examples of controllers which are external tothe flash array they control include STMicroelectrocincs's ST7265xmicrocontroller family, STMicroelectrocincs's ST72681 microcontroller,and SMSC's USB97C242, Traspan Technologies' TS-4811, ChipsbankCBM2090/CBM1190. Example of commercial IP software for Flash filesystems are: Denali's Spectra™ NAND Flash File System, Aarsan's NANDFlash Controller IP Core and Arasan's NAND Flash File System. It isappreciated that the flash controller apparatus need not be NAND-typeand can alternatively, for example, be NOR-type or phase changememory-type.

-   -   It will also be understood that the system according to the        invention may be a suitably programmed computer. Likewise, the        invention contemplates a computer program being readable by a        computer for executing the method of the invention. The        invention further contemplates a machine-readable memory        tangibly embodying a program of instructions executable by the        machine for executing the method of the invention.

Flash controlling apparatus, whether external or internal to thecontrolled flash array, typically includes the following components: aMemory Management/File system, a NAND interface (or other flash memoryarray interface), a Host Interface (USB, SD or other), error correctioncircuitry (ECC) typically comprising an Encoder and matching decoder,and a control system managing all of the above.

The present invention may for example interface with or modify, as perany of the embodiments described herein, one, some or all of the abovecomponents and particularly with the ECC component.

Features of the present invention which are described in the context ofseparate embodiments may also be provided in combination in a singleembodiment. Conversely, features of the invention, including methodsteps, which are described for brevity in the context of a singleembodiment or in a certain order may be provided separately or in anysuitable subcombination or in a different order. “e.g.” is used hereinin the sense of a specific example which is not intended to be limiting.

We claim:
 1. A method for operating a flash device, said methodcomprising: using Lee distance based codes in said flash device; therebyincreasing the number of errors that can be corrected for a given numberof redundancy cells compared with Hamming distance based codes.
 2. Amethod for operating a flash device, said method comprising: providing aLee code associated with a first group of symbols; and programming datain the flash device using a second group of symbols, said second groupbeing a sub group of the first group so as to allow extending the numberof errors that can be corrected for a given number of redundancy cellscompared with Hamming distance based codes.
 3. The method of claim 2wherein said first data stream is associated with a first group ofsymbols and said redundancy data stream is associated with a secondgroup of symbols, said second group being a sub group of the firstgroup.
 4. A decoding system comprising: a Lee metric-based decoderoperative to decode multi-level physical media.
 5. A method for errorcorrection comprising: encoding data capable of being decoded, therebyto effect error correction, said encoding comprising using a separateencoder to store a Syndrome computation to be used during said decoding.6. A system according to claim 4 wherein said multi-level physical mediaincludes media having less levels than symbol values.
 7. A method forerror correction comprising: encoding data capable of being decoded,thereby to effect error correction, said encoding comprising: encoding asyndrome giving rise to encoded syndrome; and encapsulating and mappingthe encoded syndrome into designated media cells.
 8. A method for errorcorrection comprising: encoding data capable of being decoded, therebyto effect error correction, said encoding comprising: encoding asyndrome; and reducing error probability in the encoded Syndrome byencapsulating and mapping the encoded Syndrome mixed with data bits intodesignated media cells.
 9. A storage system comprising: a flash memorycell array; a Lee distance based encoder associated with a first groupof symbols; and a writing module for programming data in at least onecell of the cell array using a second group of symbols, said secondgroup being a sub group of the first group so as to allow extending thenumber of errors that can be corrected for a given number of redundancycells compared with Hamming distance based codes.
 10. A storage systemcomprising: a flash memory cell array; and a Lee distance based encoder;wherein said Lee distance based encoder is used for error correctionencoding thereby increasing the number of errors that can be correctedfor a given number of redundancy cells compared with Hamming distancebased codes.
 11. A program storage device readable by machine, tangiblyembodying a program of instructions executable by the machine to performmethod steps for using Lee distance based codes in said flash device;thereby increasing the number of errors that can be corrected for agiven number of redundancy cells compared with Hamming distance basedcodes.